Repo link and results - https://github.com/Abhinand20/MathFormer Task: Given a factorized expression like (7-3*z)*(-5*z-9), predict the expanded form -> 15*z\*2-8\*z-63 Key takeaway: A tiny (4M param) seq2seq model trained with no math knowledge reaches ~98.6% accuracy on symbolic math tasks, suggesting it learns structural token transformations rather than any notion of operators or variables. Scaling this up could help explain why LLMs appear to “reason” mathematically, when they may actually be performing large-scale structured pattern completion. How does RL change this paradigm given the inherent architecture is still based on attention? submitted by /u/AlphaCode1 [link] [Kommentare]
A few weeks back I posted about a little experiment: a cryptocurrency that runs entirely inside a browser tab called BrowserCoin. Open a page, you're a full node. Mining is memory hard so a phone and a server farm have about the same odds. Bitcoin shaped rules, fully open source, no market, just a project to have fun with the technology. A lot of people tried it out, and I would like to recap what happened so far, because it was pretty interesting. A very important thing about this project: it's not selling anything. The coin was made to have fun with the technology. There is no market, you can't buy coins, you can't sell them. Coins exist because someone mined them. That's it. This is what the tool looks like these days: Browser Coin Landing Page Okay. What's happened, roughly in order: A wave of you showed up. The first post sent a ton of people in. At peak times there were 600+ people mining. People started building. Mini games, a faucet, mining pools, little tools and toys I had nothing to do with. Watching strangers build on something was very cool to see. Community made Projects Someone fixed my biggest weakness. In a browser you can't connect to peers by raw IP, so you need a couple of helper servers to introduce new tabs to the network, and I was running the defaults, which made me a soft point of failure for joining. A community member wrote a way better system: helpers are now discovered automatically, anyone can run one, no approval from me. They're just bootstrap hints. Your browser still validates every block locally, so a helper can't forge anything or touch your coins. That was the last slightly centralized bit, and it wasn't even me who killed it. Advertise Helper Server Feature It became more decentralized and way less killable by attacking a single server. The whole chain now backs itself up to a public repo every hour, re-verifying everything from scratch each time. If every server and tab vanished overnight, anyone could bring the network back in about fifteen minutes. Every clone is another permanent, verifiable copy. Quality of life. Faster wallet loading, a better block explorer, and a pop-out miner, a little floating window with live hashrate and CPU sliders so you can keep mining while you do other stuff. All cool features that were developed while the chain was up and running. I burned ~195k BRC (those were mined when it was mostly just me testing the chain, so I thought it made sense to burn them) to a dead address. Gone forever, no key exists, tagged in the explorer. The address is 000000000000000000000000000000000000000000000000000000000000dead, which is kinda fun. Top holder with burn Wallet People are still out here mining, and the next big challenge is doing the first hard fork. The next step is adding scripts. BrowserCoin now has a scripting engine: lock coins behind a secret and a key, a raw-script mode, a full opcode reference, and lock/redeem transactions decoded and explained in the explorer. Same chain, your balance and history carry straight over. It activates automatically off the chain's own clock on July 5, so every updated tab flips at the same block, just refresh before then. Scripting Countdown Banner Once scripts are added there's a lot of other cool stuff to experiment with. A network of strangers who don't trust each other agreeing on a shared ledger with no one in charge, using nothing but math and electricity, and now a community building games and faucets on top of it. Still the most fun thing I've made. No nonsense. So far it was very fun. No one lost anything, no one gained anything but knowledge. An experiment to have fun with the technology the way it should be. submitted by /u/swompythesecond [link] [Kommentare]
Research from 1.6M mobile sessions shows API errors drove 17.5% permanent user loss, making ambiguity worse than crashes for app growth.
Architectural Conformance for AI code — notes, benchmarks and arguments.
A. D. Aleksandrov, A. N. Kolmogorov, and M. A. Lavrent'ev Publisher: Dover PublicationsPublication Date: 1999Number of Pages: 372Format: PaperbackPrice: 38.95ISBN: 9780486409163Category: AnthologyBLL Rating: BLL**The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries. MAA Review Table of Contents [Reviewed by Michael Berg, on 08/16/2010]Mathematics: Its Content, Methods, and Meaning (Three Volumes Bound as One [!]), by (Can you believe it?) Aleksandrov, Kolmogorov, and Lavrent’ev, is a titanic work, weighing in at 356 + 374 + 351 = 1081 pages. In Part I (in two parts) a it follows a trajectory from general themes in analysis to analytic geometry, the theory of algebraic equations, and ODE. In Part II (also in two parts) we go from PDE through differential geometry, calculus of variations, and complex analysis, then through some prime number theory, probability, to (the perhaps somewhat dissonant theme of) “electronic and computing machines.” Finally, in Part III (yes, in two parts), the parth goes from real variables through linear algebra, non-Euclidean geometry, to (and this is by itself worth the price of admission:) topology (by Aleksandrov), functional analysis (by Gel’fand), and groups and then some (by Mal’cev). Well, the whole book is peppered with Russian (or Soviet — see below) luminaries: Delone, Sobolev, Ladyzenskaya, Postnikov, and Faddeev, to name a few (in addition to the aforementioned). Thus, the book under review is a compendium of rather beefy survey articles written by long-ball hitters in their fields, playing in the Soviet Union in the late 1950s, well before the empire crumbled. So it is that in the Preface to the Russian Edition the editors state that “the authors have kept in mind the goal of acquainting a sufficiently wide circle of the Soviet intelligentsia with the various mathematical disciplines, their content and methods, the foundations on which they are based, and the paths along which they have developed.” As such it covers a load of serious mathematics, accessible to, say, a strong senior undergraduate with adequate Sitzfleisch — well, presumably the reader can pick and choose: the sections are autonomous and the book’s sections can easily be read excursionally, to coin a phrase. (About the intended audience of Soviet intelligentsia, is it possible to ignore the image of Krushschev or Kosygin curling up before a warm fire in their dacha’s living room, at the end of another rough day at the Politburo, with a well-thumbed copy of the book under review in hand and a bottle of vodka at the ready? I’m willing to believe the part about the vodka …) In any event, Party leaders and apparatchiks aside, as far as the book proper is concerned, my favorite part is Part III, where, on, page 30, a cool discussion by Steckin of the Lebesgue integral is given, replete with the y-axis being sliced up, and where (on pp. 227–261) we find I. M. Gel’fand’s essay on functional analysis. Even in a treatment pitched at the present level, the master’s artistic touch cannot be hidden. Perhaps it is precisely under such circumstances that mastery shines through most remarkably. En passant, speaking of Gel’fand’s mastery, I’d recommend to the reader the gorgeous book by Gel’fand, Graev, and Piatetskii-Shapiro, Representation Theory and Automorphic Forms, just for sheer elegance. Of course, Gel’fand eventually left Russia, as the thaw set in, spending the last decades of his life at Rutgers, and Piatetskii-Shapiro eventually left for Yale and Tel-Aviv. They both died free men in 2009, Gel’fand at 96, Piatetskii-Shapiro at almost 80; God rest their souls. With both of them also being Jews, I am reminded of my own dear senior colleague, Lev Abolnikov, who is also an expatriate Russian Jew. Lev escaped from the USSR many decades ago, when the darkness was severe, and he has horror stories to tell of the regime. He tells me, also, that scientists were obliged in those days to start off their publications with special encomia to the Party and dialectical materialism. And, to be sure, we find on page iii of the book under review the ideological allusion that “[the] abstract character of mathematics gave birth even in antiquity to idealistic notions about its independence of the material world.” Even if that reads as something of an anachronism today, to read anything resembling a paean to the regime that gave us the Gulag is, to put it mildly, jarring. But turning the page, or turning over a new leaf, as one prays Russia has done too, takes us safely to Mathematics proper, and the ensuing thousand pages are, by and large, a treat and a marvelous achievement: even professionals will find a lot in these pages to enjoy and to learn. The authors have provided lists at the end of chapters of suggested further reading: an autodidact’s dream. Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA. Volume 1. Part 1 Chapter 1. A general view of mathematics (A.D. Aleksandrov) 1. The characteristic features of mathematics 2. Arithmetic 3. Geometry 4. Arithmetic and geometry 5. The age of elementary mathematics 6. Mathematics of variable magnitudes 7. Contemporary mathematics Suggested reading Chapter 2. Analysis (M.A. Lavrent'ev and S.M. Nikol'skii) 1. Introduction 2. Function 3. Limits 4. Continuous functions 5. Derivative 6. Rules for differentiation 7. Maximum and minimum; investigation of the graphs of functions 8. Increment and differential of a function 9. Taylor's formula 10. Integral 11. Indefinite integrals; the technique of integration 12. Functions of several variables 13. Generalizations of the concept of integral 14. Series Suggested reading Part 2. Chapter 3. Analytic Geometry (B. N. Delone) 1. Introduction 2. Descartes' two fundamental concepts 3. Elementary problems 4. Discussion of curves represented by first- and second-degree equations 5. Descartes' method of solving third- and fourth-degree algebraic equations 6. Newton's general theory of diameters 7. Ellipse, hyperbola, and parabola 8. The reduction of the general second-degree equation to canonical form 9. The representation of forces, velocities, and accelerations by triples of numbers; theory of vectors 10. Analytic geometry in space; equations of a surface in space and equations of a curve 11. Affine and orthogonal transformations 12. Theory of invariants 13. Projective geometry 14. Lorentz transformations Conclusions; Suggested reading Chapter 4. Algebra: Theory of algebraic equations (B. N. Delone) 1. Introduction 2. Algebraic solution of an equation 3. The fundamental theorem of algebra 4. Investigation of the distribution of the roots of a polynomial on the complex plane 5. Approximate calculation of roots Suggested reading Chapter 5. Ordinary differential equations (I. G. Petrovskii) 1. Introduction 2. Linear differential equations with constant coefficients 3. Some general remarks on the formation and solution of differential equations 4. Geometric interpretation of the problem of integrating differential equations; generalization of the problem 5. Existence and uniqueness of the solution of a differential equation; approximate solution of equations 6. Singular points 7. Qualitative theory of ordinary differential equations Suggested re Volume 2 Part 3 Chapter 6. Partial differential equations (S. L. Sobolev and O. A. Ladyzenskaja) 1. Introduction 2. The simplest equations of mathematical physics 3. Initial-value and boundary-value problems; uniqueness of a solution 4. The propagation of waves 5. Methods of constructing solutions 6. Generalized solutions Suggested reading Chapter 7. Curves and surfaces (A. D. Aleksandrov) 1. Topics and methods in the theory of curves and surfaces 2. The theory of curves 3. Basic concepts in the theory of surfaces 4. Intrinsic geometry and deformation of surfaces 5. New Developments in the theory of curves and surfaces Suggested reading Chapter 8. The calculus of variations (V. I. Krylov) 1. Introduction 2. The differential equations of the calculus of variations 3. Methods of approximate solution of problems in the calculus of variations Suggested reading Chapter 9. Functions of a complex variable (M. V. Keldys) 1. Complex numbers and functions of a complex variable 2. The connection between functions of a complex variable and the problems of mathematical physics 3. The connection of functions of a complex variable with geometry 4. The line integral; Cauchy's formula and its corollaries 5. Uniqueness properties and analytic continuation 6. Conclusion Suggested reading Part 4. Chapter 10. Prime numbers (K. K. Mardzanisvili and A. B. Postnikov) 1. The study of the theory of numbers 2. The investigation of problems concerning prime numbers 3. Chebyshev's method 4. Vinogradov's method 5. Decomposition of integers into the sum of two squares; complex integers Suggested reading Chapter 11. The theory of probability (A. N. Kolmogorov) 1. The laws of probability 2. The axioms and basic formulas of the elementary theory of probability 3. The law of large numbers and limit theorems 4. Further remarks on the basic concepts of the theory of probability 5. Deterministic and random processes 6. Random processes of Markov type Suggested reading Chapter 12. Approximations of functions (S. M. Nikol'skii) 1. Introduction 2. Interpolation polynomials 3. Approximation of definite integrals 4. The Chebyshev concept of best uniform approximation 5. The Chebyshev polynomials deviating least from zero 6. The theorem of Weierstrass; the best approximation to a function as related to its properties of differentiability 7. Fourier series 8. Approximation in the sense of the mean square Suggested reading Chapter 13. Approximation methods and computing techniques (V. I. Kr 1. Approximation and numerical methods 2. The simplest auxiliary means of computation Suggested reading Chapter 14. Electronic computing machines (S. A. Lebedev and L. V. Kantorovich) 1. Purposes and basic principles of the operation of electronic computers 2. Programming and coding for high-speed electronic machines 3. Technical principles of the various units of a high-speed computing machine 4. Prospects for the development and use of electronic computing machines Suggested reading Volume 3. Part 5. Chapter 15. Theory of functions of a real variable (S. B. Stechkin) 1. Introduction 2. Sets 3. Real Numbers 4. Point sets 5. Measure of sets 6. The Lebesque integral Suggested reading Chapter 16. Linear algebra (D. K. Faddeev) 1. The scope of linear algebra and its apparatus 2. Linear spaces 3. Systems of linear equations 4. Linear transformations 5. Quadratic forms 6. Functions of matrices and some of their applications Suggested reading Chapter 17. Non-Euclidean geometry (A. D. Aleksandrov) 1. History of Euclid's postulate 2. The solution of Lobachevskii 3. Lobachevskii geometry 4. The real meaning of Lobachevskii geometry 5. The axioms of geometry; their verification in the present case 6. Separation of independent geometric theories from Euclidean geometry 7. Many-dimensional spaces 8. Generalization of the scope of geometry 9. Riemannian geometry 10. Abstract geometry and the real space Suggested reading Part 6. Chapter 18. Topology (P. S. Aleksandrov) 1. The object of topology 2. Surfaces 3. Manifolds 4. The combinatorial method 5. Vector fields 6. The development of topology 7. Metric and topological space Suggested reading Chapter 19. Functional analysis (I. M. Gelfand) 1. n-dimensional space 2. Hilbert space (Infinite-dimensional space)< 4. Integral equations 5. Linear operators and further developments of functional analysis Suggested reading Chapter 20. Groups and other algebraic systems (A. I. Malcev) 1. Introduction 2. Symmetry and transformations 3. Groups of transformations 4. Fedorov groups (crystallographic groups) 5. Galois groups 6. Fundamental concepts of the general theory of groups 7. Continuous groups 8. Fundamental groups 9. Representations and characters of g 10. The general theory of groups 11. Hypercomplex numbers 12. Associative algebras 13. Lie algebras 14. Rings 15. Lattices 16. Other algebraic systems Suggested reading Index Tags: Surveys of Mathematics Log in to post comments Dummy View - NOT TO BE DELETED Get Ready: Our Brand New Website is Coming Soon! 2024 MAA Awards & Prize Winners Announced! Register for our OPEN Math Summer Workshops Register for MathFest 2024!
Every crash is full of people saying "buy the dip." But when the candles are actually red, most people disappear. So let's be honest: 💰 Are you buying? 🟢 Already all in? ⏳ Waiting for lower prices? 🔴 Sold and staying out? No judgment—I'm just curious what everyone's real strategy is this time. submitted by /u/Present_Let2487 [link] [Kommentare]
Turn sheet music pages using blink detection via webcam.
JavaScript, math, maps, etc
The Formula: Scale Up. Flood In. Starve Out.